Councilor at Large John R. Connolly, an attorney and former teacher, vows to shake up the Boston school system, ousting dozens from the “dysfunctional” bureaucracy and hiring an unconventional superintendent who is not a bureaucrat.

If teachers union negotiations deadlocked, Connolly said, he would simply implement the city’s final best offer.

By contrast, state Representative Martin J. Walsh, a longtime labor leader, says he would stress teamwork, partnering with the union to overhaul schools. He would pick a collaborative superintendent who would work with, not battle, the union and administrators.

The way Connolly and Walsh would execute their overhauls distinguishes the two men, who agree on many other educational issues, such as the need for a longer school day and expanded early education.

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Whether one approach or the other will lead to a more robust revitalization of Boston’s long-struggling school system is up for debate. Would Connolly’s aggressive take-charge agenda lead to chaos across the system? Would Walsh’s congeniality prevent the enactment of bold ideas?

The election is also conjuring up a much broader question: How much impact, after all, can a mayor have on bolstering the achievement of thousands of students in a large urban system like Boston’s, where some problems are seen as intractable and where schools must deal daily with the consequences of poverty.

But academic experts say that mayors might be in the best position to help failing schools raise their performance because mayors can coordinate school intervention with other measures to address students’ home lives, such as creating more affordable housing and cracking down on violent crime and drug abuse.

Bolstering the quality of the Boston school system “is the one big unsolved problem in a city where so many things are going well,” said Paul Grogan, president of the Boston Foundation.

That problem persists, Grogan said, in spite of extraordinary efforts over the past 20 years by Mayor Thomas M. Menino, who famously challenged residents in a 1996 speech to “judge me harshly” if he failed to make the city’s schools better.

The uneven caliber of schools across the city frustrates parents, even some of those whose children are in high-performing schools.

“My family has been lucky with the schools they had, and I want all families to have that kind of experience,” said Megan Wolf, of Jamaica Plain, whose children attend Boston Latin School and who is a member of QUEST, a grass-roots parent organization. “You walk into some schools and it is devastating. You see a lack of resources, like a lack of up-to-date textbooks, materials for art, or healthy food.”

In Boston, the mayor has wielded considerable control over the school system for more than two decades, with the power to appoint the School Committee and to influence the hiring of a superintendent. The effort has been credited with bringing stability to a once-tumultuous system and fostering incremental gains in student achievement.

But a national report in March by the Center for American Progress sounded a warning bell. While the report found strong gains in math and reading on national tests in Boston during the initial period of mayoral oversight, it noted that performance had tapered off in recent years.

“Success may be time-bound,” wrote the authors, Kenneth K. Wong, a Brown University education professor, and Francis X. Shen, an associate professor at the University of Minnesota Law School. “Reinventing mayoral control — whether through new leadership or new governance practices — seems necessary.’’

Whoever is elected will face enormous challenges. More than half of the city’s 127 schools rank in the bottom 20 percent in the state, primarily because of low MCAS scores. Some popular schools, such as Mission Hill K-8, Edwards Middle School and Another Course to College, had their academic standings downgraded by the state this fall.

Two low-achieving elementary schools, the Dever and Holland, face what could be the state’s first ever takeover of a Boston public school.

The high school dropout rate, which plummeted during the recent recession to below 6 percent, is now inching upward.

Many school buildings are deteriorating, while substantial gaps in achievement between students of different races and income levels persist.

But there are bright spots. Student achievement is soaring at a handful of once low-performing schools, such as Trotter Elementary, Orchard Gardens K-8 and Clap Innovation. Last spring, scores on the 10th-grade MCAS reached their highest levels, and arts and music programs are flourishing in schools across the city.

Both Connolly and Walsh say they are up to tackling the challenges.

“I think we are uniquely poised to succeed,” Connolly said in a recent interview. “We are a relatively small district and a city with vast institutional resources — colleges, universities, and nonprofits — that other cities can only dream about. We just need to be really bold in pushing changes.”

Walsh spoke of the need of building partnerships.

“We can put all the great programs in place, but we still have the challenges of what the families face,” he said. “We have a plan for the classroom, but we also have a public-health policy plan that will provide wraparound services,” such as programs to address students’ social, emotional, and physical well-being.

Paul Reville, the state’s former education secretary, believes mayors can bring about great change by their selection of leaders and by seizing the bully pulpit to promote a singular vision for public education.

“One thing Menino did well was to say how important education is and how he wanted to be held accountable for the performance of the schools,” Reville said. “He would often say, ‘If you care about the city, you have to care about the schools and these kids.’’’

As hard as it might be for many residents and parents to believe, the state of the Boston schools was worse in the early 1990s, before Raymond Flynn, the former mayor, gained the ability to appoint School Committee members. At the time, the state education commissioner was contemplating a takeover of the entire system.

Under the 13-member elected School Committee, the system ran as its own government entity and was plagued with cronyism and low student achievement. Racial tensions flared in 1990, after the committee voted along racial lines to fire the city’s first black superintendent, Laval S. Wilson.

“Everyone agreed the schools were failing,” Neil Sullivan, a former Flynn adviser who is now executive director of the Boston Private Industry Council, said in a recent interview. “The mayor would say the schools were failing because the School Committee made bad decisions. The School Committee would say the schools are failing because the city wouldn’t give them money. This would play out over and over.”

Despite its myriad problems today, the Boston school system is considered to be among the best large urban systems in the nation.

“Compared to D.C., Detroit, Chicago, and Cleveland, the Boston schools look pretty good,” said Paul Peterson, a professor at Harvard’s Kennedy School of Government.

But many Boston families yearn for a system that can offer the caliber of opportunities found in suburban schools.

The gap in performance remains wide. On the 10th-grade MCAS, 79 percent of Boston students scored proficient or higher compared to 91 percent statewide, and 64 percent scored proficient or higher in math, compared to 80 percent statewide.

Academic experts say gaps will always exist in a district where about three-quarters of its students come from low-income households, but they say mayors should not shy away from pursuing measures that would help more students succeed.